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Asymptotic iteration method for eigenvalue problems. (English) Zbl 1070.34113
The main task of the present work is to introduce a new technique to solve second-order homogeneous linear differential equations $$ y''=\lambda_0(x)y'+s_0(x)y,\tag* $$ where $\lambda_0,s_0\in C_\infty(a,b)$. The following result is established: The differential equation (*) has the general solution $$ y(x)=\exp\left(-\int^x\alpha dt\right)\left[C_2+C_1\int^x\exp\left(\int^t(\lambda_0(\tau)+2\alpha(\tau))d\tau\right)dt\right] $$ if for some $n>0$ $$ \frac{s_n}{\lambda_n}=\frac{s_{n-1}}{\lambda_{n-1}}\equiv\alpha, $$ where $\lambda_k=\lambda'_{k-1}+s_{k-1}+\lambda_0\lambda_{k-1}$ and $s_k=s'_{k-1}+s_0\lambda_{k-1}$ for $k=1,2,...,n$. Applications to Schrödinger-type problems, including someones with highly singular potentials, are presented.

34L16Numerical approximation of eigenvalues and of other parts of the spectrum
34A30Linear ODE and systems, general
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis
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